Primitive ideals and nilpotent orbits in type \(G_ 2\)
type: Item
instance of: scholarly article
title: Primitive ideals and nilpotent orbits in type \(G_ 2\)
zbMATH Open document ID: 0644.17005
DOI: 10.1016/0021-8693(88)90214-1
author: Thierry Levasseur
author: S. Paul Smith
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be a finite dimensional semi-simple Lie algebra over \({\mathbb{C}}\) and J a primitive ideal of its enveloping algebra U(\({\mathfrak g})\). The associated variety \({\mathcal V}(J)\subseteq {\mathfrak g}\) * is defined as the collection of zeros of the associated graded ideal, gr J, in S(\({\mathfrak g})\), the symmetric algebra on \({\mathfrak g}\), and \({\mathcal V}(J)\) can be considered as a subvariety of \({\mathfrak g}\). Let G denote the adjoint algebraic group of \({\mathfrak g}\). If \(X\in {\mathfrak g}\) and ad X acts nilpotently on \({\mathfrak g}\), then G.X is called a nilpotent orbit. By a result of \textit{A. Joseph} [J. Algebra 93, 509-523 (1985; Zbl 0594.17009)], \({\mathcal V}(J)\) is the closure of a single nilpotent G-orbit. Now suppose \({\mathfrak g}_ 2\) is the simple Lie algebra of type \(G_ 2\). The nilpotent orbits are of dimension 0, 6, 8, 10, 12. \textit{A. Joseph} [J. Algebra 65, 284-306 (1980; Zbl 0441.17004) and Lect. Notes Math. 880, 249-266 (1981; Zbl 0468.17004)] has shown that there are precisely two completely prime primitive ideals associated with the unique nilpotent orbit of dimension 8. The paper examines this orbit and these ideals by embedding \({\mathfrak g}_ 2\) in so(7) and the Joseph ideal in U(so(7)), thus answering a number of (as yet, unpublished) questions of Vogan.
reviewed by: Patrick F. Smith
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 4049836
zbMATH Keywords: enveloping algebra
zbMATH Keywords: variety
zbMATH Keywords: simple Lie algebra of type \(G_ 2\)
zbMATH Keywords: primitive ideals
zbMATH Keywords: nilpotent orbit of dimension 8
zbMATH Keywords: Joseph ideal
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90214-1
OpenAlex ID: W2151680788
cites work: Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie- Algebra
cites work: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen
cites work: Q5580319
cites work: Q3682663
cites work: Q4416343
cites work: Constructions of the Weil representations of certain simple Lie algebras
cites work: Q4143433
cites work: Q4089001
cites work: Moduln mit einem höchsten Gewicht
cites work: Q4152693
cites work: The minimal orbit in a simple Lie algebra and its associated maximal ideal
cites work: On the annihilators of the simple subquotients of the principal series
cites work: Gelfand-Kirillov Dimension for the Annihilators of Simple Quotients of Verma Modules
cites work: Goldie rank in the enveloping algebra of a semisimple Lie algebra. I, II
cites work: Goldie rank in the enveloping algebra of a semisimple Lie algebra. III
cites work: Q3921372
cites work: On the variety of a highest weight module
cites work: On the associated variety of a primitive ideal
cites work: La dimension de Krull de U(s\(\ell (3))\). (The Krull dimension of U(s\(\ell (3)))\)
cites work: Q4767406
cites work: On the fixed point set of a unipotent element on the variety of Borel subgroups
cites work: On the desingularization of the unipotent variety
cites work: ON A CLASS OF QUASIHOMOGENEOUS AFFINE VARIETIES
cites work: Q3710659
description: scientific article; zbMATH DE number 4049836